Nine prisoners are taken for their daily exercise handcuffed together in
threes. How would the warden arrange the men each day so that no two men
are handcuffed together more than once over a six day period?

I was working on calculating how many combinations of pizza can be
made using 6 different toppings with no double toppings allowed, and I
found that both c(6,0) + c(6,1) + c(6,2) + ... + c(6,6) and 2^6 gave
me the same correct answer of 64. Why do both methods work?

How many different ways can you arrange the elements in a set? Two
elements (a and b) can be ab or ba, but it's harder with bigger sets.
An explanation of the math underlying the Multiplication Principle.

I know how to use the formula to determine the number of combinations
of 6 numbers drawn from 12 numbers (1 to 12 inclusive). But without
the formula, how can I make a list of all the possible combinations
and be confident that I've found them all?

We are to record 512 different combinations of each level (or color) of
rod, with the orange rod (the largest) being the last. Is there a formula
exist to find these combinations without literally manipulating the rods?